Varieties of Distributivity: One by One vs Each
Adrian Brasoveanu, Robert Henderson – UC Santa Cruz
abrsvn,[email protected]
SALT 19 – Ohio State University, April 5, 2009
1
Introduction
(4)
Example (1) is compatible with events that unfold in very different ways. The
addition of one by one in (2) constrains how the leaving events can proceed.
(1)
The boys left.
(2)
The boys left one by one.
(5)
The boys each recited a poem.
G
g1
g2
...
• it breaks an event down into temporally sequenced subevents
• it distributes the plural participant over these subevents
=
e1
leave(e1 )
runtime(e1 )
(3)
the.boys =
boy1
ag(e1 ) = boy1
⊕
e2
leave(e2 )
≺ runtime(e2 )
⊕
boy2
ag(e2 ) = boy2
y (poems)
poem1 (= g1 (y))
poem2 (= g2 (y))
...
boy1 recited poem1
boy2 recited poem2
...
∗ each breaks the plural individual the.boys into atoms and stores
each boy in a variable assignment: boy1 in g1 , boy2 in g2 etc.
∗ the remainder of the sentence is interpreted relative to each variable assignment, so we store a (possibly different) poem in each
assignment, relative to each boy
Intuitively, one by one is an event modifier that targets a plural participant in
the event.
e
x (boys)
boy1 (= g1 (x))
boy2 (= g2 (x))
...
– encapsulation into a function: functions, e.g., Skolem functions,
store quantificational dependencies as a whole, mapping each entity
to the (possibly non-atomic) entity that depends on it (Stone 1999,
Bittner & Trondhjem 2008 and Dekker 2008 among others)
⊕ ...
...
≺ ...
⊕ ...
...
(6)
The boys recited a poem one by one.
x (boys)
Goals:
(7)
• immediate goal (on which we focus today): investigate the constraints on the distribution and interpretation of one by one and give a
compositional semantics for one by one that captures them
g
the.boys
f (boy-poem dependency)


boy1 7→ poem1
f = boy2 7→ poem2 
...
Both encapsulation and decomposition allow cross-sentential anaphora (see
Stone 1999 and Bittner & Trondhjem 2008 for encapsulation-based accounts
of such anaphora).
• larger goal: motivate two routes to establishing distributive quantificational dependencies – exemplified by each and one by one, respectively
– decomposition into sets of assignments: each n-tuple of quantificationally dependent entities is individually stored in a variable
assignment and quantifiers are interpreted relative to the entire set
of variable assignments (van den Berg 1996 and Nouwen 2003 among
others)
1
(8)
a. One by one, the boys chose a book.
b. Then, one by one, they opened it and read out the title.
(9)
a. The boys each chose a book.
b. Then, they each opened it and read out the title.
• one by one tokens: 2774; frequency: ∼ 7 / mil (spoken: 4 / mil, fiction
19.7 / mil, magazines: 6.1 / mil, newspapers 4.3 / mil, academic 2.1 / mil)
The two routes to distributivity enable us to capture the fact that each, but
not one by one, licenses internal readings of different:
(10)
The boys each recited a different poem.
Main generalizations:
• for any two boys a and b, the poem a recited is different from the poem b
recited
• one by one needs to target a nominal argument / adjunct
• the nominal argument / adjunct has to be local – basically, in the same
clause as the verbal predicate that one by one modifies
• in (5), we get this by comparing variable assignments in a pairwise way
(see Brasoveanu 2008 for more details)
(11)
2.1
The boys recited a different poem one by one.
Types of Nominal Targets
• subjects
• the internal reading is not available because there is only one variable
assignment in (7)
(12)
Core proposal: one by one is an event modifier that encapsulates part of a
θ-role function.
One by one, 63 North Koreans stepped through the heavily fortified
border zone.
(13)
The sucker holes closed, one by one.
• direct objects
• θ-role functions map events to their participants
• one by one targets a θ-role function and structures it so that linearly ordered atomic subparts of the event are mapped onto atomic subparts of a
plural participant
One by one, he eliminates the contestants.
(15)
I brought him my singers one by one.
• prepositional phrases
• so, one by one does not directly introduce the boy-poem function f in (7)
above – such a function is induced via events and θ-role functions
Roadmap:
• the distribution and interpretation of one by one
• outline of the account: one by one and θ-roles
(16)
Joe came back for the boxes, one by one.
(17)
He waved forked fluorescent scanners over the vials one by one to catch
anything in the solutions that didn’t register chemically.
(18)
He was followed, one by one, by his companions.
However, there seem to be some restrictions on the kind of nominals that one
by one can target.
• conclusion
2
(14)
The Distribution of One by One
Most of the examples in this section are from the Corpus of Contemporary
American English (http://www.americancorpus.org/).
• large, balanced corpus of contemporary American English – 385 million
words of text, including 20 million words each year from 1990-2008, equally
divided among spoken (78.8 million words), fiction (74.9 million words),
popular magazines (80.7 million words), newspapers (76.3 million words)
and academic texts (76.2 million words)
(19)
I loaded pumpkins onto the truck one by one.
(20)
*I loaded the truck with pumpkins one by one.
(21)
I sprayed the walls with (a can of) paint one by one.
(22)
*I sprayed (a can of) paint onto the walls one by one.
One by one targets a variety of arguments / adjuncts – but crucially, it must
have a target. Out of 1000 occurrences in COCA that we examined closely,
there is only one more or less felicitous example without a nominal target.
2
(23)
[Inside an old factory building in Cambridge, Mass., a remarkable machine
with the improbable name Zeus is hard at work. Flexing its two robotic arms,
the computer-driven device reaches again and again into a storage area the
size of a toddler’s crib, where thousands of individual samples of genetic material sit in tiny wells etched into plastic plates, each one identified by a unique
bar code.]
One by one, Zeus searches for a particular code, dips into the corresponding well with a fine, quill-like probe and picks up a minuscule
droplet of liquid DNA.
(24)
(34)
*But when their units were activated, Mr. Lozano slowly realized that
if war in the Persian Gulf came one by one, his family could be wiped
out.
(35)
But when their units were activated, Mr. Lozano slowly realized that
if war in the Persian Gulf came, his family could be wiped out one by
one.
(wipe out one by one)
One by one can operate across multiple, conjoined predicates only when it
syntactically scopes over them.
[Compare with:] Well by well, Zeus searches for a particular code.
The plural-nominal-target constraint distinguishes one by one from other similar event modifiers, e.g., brick by brick, simultaneously or consecutively.
(36)
Over those busy decades, as one by one our nestlings fledged and took
wing . . .
(Xfledge one by one)
(37)
Over those busy decades, as our nestlings fledged and one by one took
wing . . .
(*fledge one by one)
(25)
I tore the house down brick by brick.
(26)
*I tore the house down one by one.
(plurality required)
(27)
The city ordered the demolition to be done brick by brick.
(38)
(28)
*The city ordered the demolition to be done one by one.
(nominal target required)
It was sad that one by one our nestlings fledged and took wing. (Xtake
wing one by one)
(39)
It was sad that one by one our nestlings fledged and that they took
wing. (*take wing one by one)
(29)
It got dark and it started to rain simultaneously.
(30)
Consecutively, it got dark and it started to rain.
(31)
*It got dark and it started to rain one by one.
(nominal target required)
2.2
Locality can be specified in syntactic terms (clause-boundedness) or in semantic
terms.
For example, we could say that one by one needs to target a participant in
the event it modifies. But this event can have a complex internal structure –
e.g., make regret in (40) below – and one by one can target a participant in an
embedded subevent.
Locality of the Nominal Target
One by one does not require strict adjacency, but the nominal target has to be
in the same clause as the event-contributing predicate that one by one modifies.
(40)
If causative structures like make regret are syntactically biclausal, then characterizing the locality of one by one in semantic terms is more adequate. For
simplicity, we will continue to characterize it in syntactic terms.
• the felicity of the sentences below depends on the position of one by one
• we obtain different readings depending on the position of one by one
(32)
(33)
One by one, I’m going to make Harvey regret all those lies he told.1
But when, one by one, their units were activated, Mr. Lozano slowly
realized that if war in the Persian Gulf came his family could be wiped
out.
(activate one by one)
2.3
The Internal Structure of the Nominal Target
The plurality of the nominal target does not need to be morphologically realized.
One by one is compatible with:
*But one by one, when their units were activated, Mr. Lozano slowly
realized that if war in the Persian Gulf came his family could be wiped
out.
1 We
3
are indebted to Jorge Hankamer for this example.
3
• group-denoting nouns
(41)
One by one that baffled, costumed crew slunk away into the shadows.
(42)
She said good-bye to her staff one by one.
• syntactically: one by one is a verbal adjunct (Jackendoff 2008)
• conjoined NPs
(43)
(44)
• semantically: one by one is an event modifier that targets a plural participant in the event it modifies
Jan dredged from a Safeway bag, one by one, a can of baby corn cobs,
a tin of Norwegian sardines, and a glass jar crammed with tiny white
cocktail onions.
• not immediately obvious how to compositionally capture the fact that one
by one simultaneously targets an event and an individual
In the next hour, a manager, a lawyer and three publicists will, one by
one, approach Ms. Paltrow’s table.
Proposal: one by one targets the cumulative closure of a θ-role function.
• we capture the nominal target constraint: θ-roles are not assigned when
there is no overt argument
• quantifiers headed by every / each
(45)
One by one, every student present began to applaud.
(46)
The provost called each man in the company one by one to be interviewed.
(47)
One by one, we each set our Ortovox transceivers on transmit while the
others listen.
(48)
One by one each tells his story of life around the King of Kings.
Outline of the Account: One by One and θrole Encapsulation
• we capture locality: θ-roles are clause-bounded
• the cumulative closure enables us to account for nominal targets that are
morphologically singular, but semantically plural, e.g., DPs headed by every / each
Cumulative closure:
• pseudo-partitives
(49)
A squad of unknown terrorists walked one by one into several subway
stations during the peak of rush hour Friday afternoon.
(50)
One by one, he dropped on the table a series of snapshots he had taken.
– properties: for any set P , *P is the smallest set such that P ⊆ *P
and, if a ∈ *P and b ∈ *P , then a ⊕ b ∈ *P (Link 1983)
– relations (or functions): for any n-place relation R, **R is the smallest relation such that R ⊆ **R and, if ha1 , . . . , an i ∈ **R and
hb1 , . . . , bn i ∈ **R, then ha1 ⊕ b1 , . . . , an ⊕ bn i ∈ **R (based on
Krifka 1986)
• partitives, including most / each-based partitives
(51)
Linguini watches – stunned – as, one by one, the rest of the staff exits.
(52)
Most of those killed were unarmed prisoners, boys and men, shot in
groups, or sometimes one by one.
(53)
Then, one by one, at several-minute intervals, each of us sets out on a
solitary walk.
Thus, the distributive quantificational dependencies introduced by one by one
are the dependencies encapsulated in the targeted θ-role function.
3.1
A Closer Look at the Formalization
We work with classical (many-sorted) type logic:
• the domain of individuals of type e is the powerset of a designated set of
entities IN minus the empty set: De = ℘+ (IN) = ℘(IN) \ ∅
Summary of the generalizations:
• one by one targets a nominal argument / adjunct
• the nominal argument / adjunct has to be local – in the same
clause as the verbal predicate that one by one modifies
• the nominal target needs to be semantically, but not morphologically, plural
• the domain of events of type is the powerset of a designated set of events
EV minus the empty set: D = ℘+ (EV) = ℘(EV) \ ∅
4
• atomic individuals and atomic events are the singleton sets in ℘+ (IN) and
℘+ (EV) respectively; they are identified by a predicate atom (which applies
to both individuals and events)
The last three conjuncts in (56) give the contribution of one by one:
• linear.order({e0 ≤ e : atom(e0 )}) requires the atomic subevents of the
event e under discussion to be temporally sequenced
• the “part of” relation ≤ over individuals / events is set inclusion over
℘+ (IN) / ℘+ (EV): a ≤ b iff a ⊆ b
• | {xe ≤ **θ(e) : atom(x)} | > 1 ensures that one by one targets only plural
participants in the event
• the sum operation ⊕ is set union over ℘+ (IN) / ℘+ (EV): a ⊕ b := a ∪ b
• ∀e0 ≤ e(atom(e0 ) → | {xe ≤ **θ(e0 ) : atom(x)} | = 1) ensures that there
is a one-to-one, θ-based correspondence between each atomic subevent of
the event e and each atomic individual in **θ(e)
• θ-roles are functions of type e from events (type ) to individuals (type e)
• for example, th is the theme role, ag is the agent role etc.
So, one by one contributes part of the dependency encapsulated in the function
θ: that part that relates the atoms of the plural event e and their (atomic)
participants.
Two by two has the same translation as one by one except that the last subformula is | {xe ≤ **θ(e0 ) : atom(x)} | = 2.
• cumulative closure for θ-role functions: **ag is the smallest set of
hevent, individuali-pairs such that ag ⊆ **ag and, if ag(e) = x and
ag(e0 ) = x0 , then **ag(e ⊕ e0 ) = x ⊕ x0
Arguments and adjuncts are event modifiers:
3.2
• they have denotations of type (t)(t)
In (57), one by one can only be indexed with the ag role because the theme is
singular.
• they have translations of the form λPt .λe . P(e) ∧ . . .
(57)
Arguments and adjuncts are syntactically indexed with their θ-roles and this
indexation percolates up the tree: as we go higher in the functional layer built
on top the main lexical verb, we accumulate more and more θ-role indices.
These θ-role indices are not visible beyond the top functional projection of the
lexical verb (hence one by one modification is clause-bounded).
(54)
[ag,th
[DP
IP
The
ag
boys ]
[th
I0
recited [DP ‘The Raven’
th
one by oneθ
1 ,θ2 ,...
[θXP
. . . ],
Theag boys recited ‘The Raven’th one by oneag .
(58) recite
λe . recite(e)
(59) ‘The Raven’th
λPt .λe .
P(e) ∧ th(e) = the-raven
(60) theag
]]]
(61) boys
(62) PAST
Syntax/semantics constraint: one by one can target any of the θ-roles that
its syntactic sister is indexed with – but only them.
(55)
Some Examples
λPet .λPt .λe . P(e) ∧ ag(e) = σx. P (x)
λxe . *boy(x)
λPt . ∃e (P(e) ∧ runtime(e) ≺ now)
(63) ∃e (recite(e) ∧ th(e) = the-raven ∧
ag(e) = σx. *boy(x) ∧
runtime(e) ≺ now ∧
linear.order({e0 ≤ e : atom(e0 )}) ∧
| {xe ≤ **ag(e) : atom(x)} | > 1 ∧
∀e0 ≤ e(atom(e0 ) → atom(**ag(e0 ))))
where θ ∈ {θ1 , θ2 , . . .}
• since one by one needs to target a θ-role, this constraint effectively requires
its sister node to be indexed with at least one θ-role
The meaning of one by one (subscripts on terms indicate their types):
e
e1
atom(e1 )
runtime(e1 )
(64)
the.boys =
boy1
atom(boy1 )
ag(e1 ) = boy1
(56) one by one θ
λPt .λe . P(e) ∧
linear.order({e0 ≤ e : atom(e0 )}) ∧
| {xe ≤ **θ(e) : atom(x)} | > 1 ∧
∀e0 ≤ e(atom(e0 ) → | {xe ≤ **θ(e0 ) : atom(x)} | = 1)
• |X| is the cardinality of X
5
=
⊕
e2
atom(e2 )
≺ runtime(e2 )
⊕
boy2
atom(boy2 )
ag(e2 ) = boy2
⊕ ...
...
≺ ...
⊕ ...
...
...
References
In contrast, (65) is ambiguous: one by one can target either the ag or the th
role because they are both associated with plural individuals.
(65)
van den Berg, M. (1996). Some Aspects of the Internal Structure of Discourse.
PhD dissertation, University of Amsterdam.
Bittner, M. (2001). Topical Referents for Individuals and Possibilities. In Proceedings of SALT 11, R. Hastings et al (eds.), Ithaca: CLC, 36-55.
Bittner, M. & Trondhjem, N. (2008). Quantification as Reference: Evidence
from Q-verbs. In Quantification: A Cross-Linguistic Perspective, L. Matthewson (ed).
Brasoveanu, A. (2008). Sentence-Internal Readings of Same/Different as
Quantifier-Internal Anaphora. In Proceedings of WCCFL 27, 72-80.
Carlson, G. (1987). Same and Different: Some Consequences for Syntax and
Semantics. In Linguistics and Philosophy 10, 531-565.
Dekker, P. (2008). A Multi-dimensional Treatment of Quantification in Extraordinary English. In Linguistics and Philosophy 31, 101-127.
Gallin, D. (1975). Intensional and Higher-Order Modal Logic with Applications
to Montague Semantics. North-Holland Mathematics Studies.
Jackendoff, R. (2008). Construction after Construction and Its Theoretical
Challenges. In Language 84, 8-28.
Krifka, M. (1986). Nominalreferenz und Zeitkonstitution: Zur Semantik von
Massentermen, Pluraltermen und Aspektklassen, PhD dissertation, University of Munich. Published by Wilhelm Finck, Munich, 1989.
Krifka, M. (1989). Nominal Reference, Temporal Constitution and Quantification in Event Semantics. Semantics and Contextual Expressions, R. Bartsch,
J. van Benthem & P. von Emde Boas (eds.), Dordrecht: Foris Publication.
Link, G. (1983). The Logical Analysis of Plural and Mass Nouns: A Latticetheoretic Approach. In Meaning, Use, and Interpretation of Language, R.
Bäuerle et al (eds), Berlin: de Gruyter, 302-323.
Nouwen, R. (2003). Plural Pronominal Anaphora in Context. PhD Dissertation,
University of Utrecht.
Stone, M. (1999). Reference to Possible Worlds, RuCCS 49.
Theag boys recited theth poems one by oneag/th .
The fact that θ-role cumulative closure is built into the meaning of one by one
allows us to account for the fact that one by one can target DPs headed by
each.
(66)
One by oneag , eachag boy recited ‘The Raven’th
For simplicity, let us give each an encapsulated treatment: each simultaneously
distributes over events and individuals.
(67) each ag
λPet .λPt .λe .
∀e0 ≤ e(atom(e0 ) → P(e0 )) ∧
ag[{e0 ≤ e : atom(e0 )}] = {xe : atom(x) ∧ P (x)}
• θ[E] = X abbreviates X = {θ(e) : e ∈ E}, i.e., the set X is the image of
the set E under the function θ
(68) ∃e (∀e0 ≤ e(atom(e0 ) →
recite(e0 ) ∧ th(e0 ) = the-raven) ∧
ag[{e0 ≤ e : atom(e0 )}] =
{xe : atom(x) ∧ boy(x)}
∧ runtime(e) ≺ now ∧
linear.order({e0 ≤ e : atom(e0 )}) ∧
| {xe ≤ **ag(e) : atom(x)} | > 1 ∧
∀e0 ≤ e(atom(e0 ) → atom(**ag(e0 ))))
4
Conclusion
• one by one is an event modifier that targets a plural participant in the
event it modifies
• one by one distributes over this participant by targeting its θ-role
• the resulting distributive dependencies are different from the distributive
dependencies introduced by items like each
Acknowledgments
A great many thanks to Pranav Anand, Scott AnderBois, Sandy Chung, Donka
Farkas, Vera Gribanova, Jorge Hankamer, a SALT reviewer and the UCSC Scircle audience for comments on this work. All mistakes remain our own.
6